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$.0030077$ radians is $\arctan(.0030077)$ degrees?

In this question Related Rates Hot Air Balloon, the op converts from radians to degrees, by taking the $\arctan(\text{radians})$. This is obviously a wrong method. In fact if your calculator is set to degrees then $\arctan$ will give you an answer totally off. But if your calculator is set to degrees it gives a good approximation for $x \approx 0^+$:

So if you just find the $\arctan$ of your radians you will get a close enough answer if your calculator is set to degrees.

I believe this has something to do with:

$$\lim_{x \to 0^+} \frac{\arctan (x)}{x}=1$$

For $x \approx 0^+$

$$\arctan (x) \approx x$$

Hence for $x$ near $0^+$:

$$\arctan (\frac{180}{\pi}x) \approx \frac{180}{\pi}x$$

In which case if your calculator assumes $x$ to be in degrees when calculating $\arctan (x)$ so it converts it to $\frac{180}{\pi}x$ then it it gives you back $\approx \frac{180}{\pi}x$ which is the expression for radians if $x$ is degrees.

Is this correct. I'm don't know much on this so I'm not trying to give any false information.

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Ahmed S. Attaalla
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    On a cursory reading, I see nothing whatever wrong with your reasoning. Indeed, it seems a serious error of category to take the arctan of an an angle, as you report . – Lubin Aug 21 '16 at 04:34

1 Answers1

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While angles can be measured in radians as well as in degrees the function $\arctan$ accepts pure real numbers (e.g., slopes of lines in the $(x,y)$-plane) as inputs and gives an angle, measured in radians as output. If $|x|\ll1$ then $\arctan x\approx x$. It follows that $$\alpha:=\arctan(0.0030077)\approx 0.0030077\ ,$$ whereby the value on the right hand side is the value of $\alpha$, expressed in radians. You can convert this angle into degrees as follows: $$\alpha={180\over\pi}\>0.0030077\ ^\circ=0.17233\>^\circ \ .$$

Christian Blatter
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